On linear projections of quadratic varieties (Q2796700)

In the article under review, the authors consider a non-degenerate irreducible projective variety \(X \subset \mathbb{P}^r\) which satisfies condition \(K_2\), in the sense of \textit{P. Vermeire} [Compositio Math. 125 No. 3, 263--282, (2001; Zbl 1056.14016)], and study the linear projection \(\pi_q : X \rightarrow \mathbb{P}^{r-1}\) determined by a point \(q \in \mathbb{P}^r\) lying outside of \(X\).NEWLINENEWLINEIn more detail, let \(X_q = \pi_q(X) \subset \mathbb{P}^{r-1}\) and NEWLINE\[NEWLINEs(q) := h^0(\mathbb{P}^r,\mathcal{I}_X(2)) - h^0(\mathbb{P}^{r-1}, \mathcal{I}_{X_q}(2))-1.NEWLINE\]NEWLINE The authors show that, when \(X\) satisfies condition \(K_2\), the geometry of \(X_q\) and the morphism \(f_q : X \rightarrow X_q\) induced by \(\pi_q\) is related to the number \(s(q)\).NEWLINENEWLINEAs some specific examples, the authors prove that \(s(q)>0\) and that the morphism \(f_q\) is birational. They also show that the secant cone \(\mathrm{Sec}_q(X) \subset \mathbb{P}^r\) is a linear subspace of dimension \(r-s(q)\) while the singular locus \(\Lambda := \mathrm {Sing}(f_q) \subset \mathbb{P}^{r-1}\), which coincides with the support of the sheaf \((f_q)_* \mathcal{O}_X / \mathcal{O}_{X_q}\), is a linear subspace of dimension \(r-s(q)-1\). The sheaf \((f_q)_* \mathcal{O}_X / \mathcal{O}_{X_q}\) is also shown to be isomorphic to \(\mathcal{O}_\Lambda(-1)\).NEWLINENEWLINEAs one consequence of their main result, the authors deduce that if, additionally, \(X\) is normal, then \(f_q : X \rightarrow X_q\) is the normalization of \(X_q\).NEWLINENEWLINEFinally, the authors illustrate their main result by considering various simple exterior projections of the rational normal scroll \(S(1,1,4) \subset \mathbb{P}^8\).